Answer:
The dimensions should be length=width=height=74 cm.
Step-by-step explanation:
We have an optimization with restriction problem.
We have to maximize the volume, subject to restriction in the sum of the length, width and height.
Let x be the length and width, that are equal, and z be the height.
The restriction can be expressed as:
[tex]x+x+z\leq222\\\\2x+z\leq222[/tex]
We can express z in function of x as:
[tex]2x+z=222\\\\z=222-2x[/tex]
The volume, the function to be optimized, can be expressed as:
[tex]V=x^2z=x^2(222-2x)=222x^2-2x^3[/tex]
To optimize, we derive and equal to zero.
[tex]\dfrac{dV}{dx}=\dfrac{d}{dx}[222x^2-2x^3]=2*222x-3*2x^2=444x-6x^2=0\\\\\\444x-6x^2=0\\\\x(444-6x)=0\\\\444-6x=0\\\\x=444/6=74[/tex]
We have the optimum length. We can now calculate the height z:
[tex]z=222-2(74)=222-148=74[/tex]
